# How to bootstrap confidence interval for large dataset

Suppose we want to calculate confidence interval of mean value. If the dataset is massive ($n$ samples), classical bootstrapping is hard to apply since the size of the resample must be $n$.

But if I resample the dataset with a smaller size $k$, the variance of the bootstrap distribution will be different from the original one. Can I simply divide it by $\frac{n}{k}$? Is there some advice to choose $k$?

Thanks.

• See Hastie and Efron's new book Computer Age Statistical Inference where these issues are dealt with in depth. – DJohnson Jan 15 '17 at 13:59
• There is a bootstrap method called m out of n bootstrap where the bootstrap sample size is m<n for n being the sample size of the original data set. Whether or not this helps you with very large n I am not sure because it requires that m/n does not go to zero with large n. so you can't conveniently take a small m and a very large n. – Michael Chernick Jan 15 '17 at 15:46
• Why do you want to estimate the variance? You state you actually want the confidence interval. – mdewey Jan 15 '17 at 16:28
• @mdewey It's actually used to evaluate an AB test, so I use a t-test after estimating the variance of each experiment group. – Morrissss Jan 16 '17 at 2:39

## 2 Answers

If your sample is large enough, you could simply use the Central Limit Theorem to derive your confidence interval of the mean.

Although the classical bootstrap requires you to resample for $n$ samples, but this is always not necessary especially if you have a large data set.

You should set $k$ to as large as you can handle computationally.

Standard error for your sample mean is defined as:

If your k ($n$ in the formula) is large enough, your sample mean should be unbiased and your confidence interval should be small enough practically.

• Thanks for reply. One more question: can I estimate the standard error of the original $n$ samples as $\frac{s \sqrt{k}}{\sqrt{n}}$, where $s$ is the SE of bootstrap distribution. – Morrissss Jan 16 '17 at 2:42
• @Morrissss I'm not sure, maybe you start a new question for it? – SmallChess Jan 17 '17 at 2:35